Revision #2 Authors: Shachar Lovett, Roy Meshulam, Alex Samorodnitsky

Accepted on: 20th October 2008 00:00

Downloads: 1277

Keywords:

Let $p$ be a fixed prime number, and $N$ be a large integer. The 'Inverse Conjecture for the Gowers norm' states that if the "$d$-th Gowers norm" of a function $f:F_p^N to F_p$ is non-negligible, that is larger than a constant independent of $N$, then $f$ can be non-trivially approximated by a degree $d-1$ polynomial. The conjecture is known to hold for $d=2,3$ and for any prime $p$. In this paper we show the conjecture to be false for $p=2$ and for $d = 4$, by presenting an explicit function whose $4$-th Gowers norm is non-negligible, but whose correlation any polynomial of degree $3$ is exponentially small. Essentially the same result (with different correlation bounds) was independently obtained by Green and Tao cite{gt07}. Their analysis uses a modification of a Ramsey-type argument of Alon and Beigel cite{ab} to show inapproximability of certain functions by low-degree polynomials. We observe that a combination of our results with the argument of Alon and Beigel implies the inverse conjecture to be false for any prime $p$, for $d = p^2$.

Revision #1 Authors: Shachar Lovett, Roy Meshulam

Accepted on: 6th May 2008 00:00

Downloads: 1253

Keywords:

Let $p$ be a fixed prime number, and $N$ be a large integer.

The 'Inverse Conjecture for the Gowers norm' states that if the

"$d$-th Gowers norm" of a function $f:F_p^N to F_p$ is

non-negligible, that is larger than a constant independent of $N$,

then $f$ can be non-trivially approximated by a degree $d-1$

polynomial. The conjecture is known to hold for $d=2,3$ and for any

prime $p$. In this paper we show the conjecture to be false for $p=2$

and for $d = 4$, by presenting an explicit function whose $4$-th

Gowers norm is non-negligible, but whose correlation any polynomial

of degree $3$ is exponentially small. Essentially the same result

(with different correlation bounds) was independently obtained by

Green and Tao cite{gt07}. Their analysis uses a modification of a

Ramsey-type argument of Alon and Beigel cite{ab} to show

inapproximability of certain functions by low-degree polynomials. We

observe that a combination of our results with the argument of Alon

and Beigel implies the inverse conjecture to be false for any prime

$p$, for $d = p^2$.

TR07-123 Authors: Shachar Lovett, Roy Meshulam, Alex Samorodnitsky

Publication: 7th December 2007 11:45

Downloads: 1233

Keywords:

Let $p$ be a fixed prime number, and $N$ be a large integer.

The 'Inverse Conjecture for the Gowers norm' states that if the "$d$-th Gowers norm" of a function $f:\F_p^N \to \F_p$ is non-negligible, that is larger than a constant independent of $N$, then $f$ can be non-trivially approximated by a degree $d-1$ polynomial. The conjecture is known to hold for $d=2,3$ and for any prime $p$. In this paper we show the conjecture to be false for $p=2$ and for $d = 4$, by presenting an explicit function whose $4$-th Gowers norm is non-negligible, but whose correlation any polynomial of degree $3$ is exponentially small.

Essentially the same result (with different correlation bounds) was independently obtained by Green and Tao \cite{gt07}. Their analysis uses a modification of a Ramsey-type argument of Alon and Beigel \cite{ab} to show inapproximability of certain functions by low-degree polynomials.

We observe that a combination of our results with the argument of Alon and Beigel implies the inverse conjecture to be false for any prime $p$, for $d = p^2$.